Have you ever wondered about the angles inside a shape or the angles formed when parallel lines are intersected by a transversal? These are known as interior angles, and they play a crucial role in geometry. In this article, we'll delve into the world of interior angles, from their types and properties to formulas and practical applications.
What are Interior Angles?
Interior angles can be defined in two ways:
- Angles inside a Polygon: When we talk about interior angles, we are often referring to the angles that lie inside a shape, typically a polygon. For example, in the figure below, angles ∠a, ∠b, and ∠c are interior angles.
- Interior Angles of Parallel Lines: Interior angles can also refer to the angles that lie in the area enclosed between two parallel lines intersected by a transversal. In the figure below, lines (L_1) and (L_2) are parallel, and L is the transversal. The angles ∠1, ∠2, ∠3, and ∠4 are interior angles.
In geometry, interior angles are categorized based on their measurements. It's important to note that there are other types of angles known as pair angles, which appear in pairs to exhibit certain properties. Interior angles fall into this category.
Types of Interior Angles
When two straight lines are cut by a transversal, two types of interior angles are formed: alternate interior angles and co-interior angles.
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Alternate Interior Angles: These angles are formed when two parallel lines are intersected by a transversal. They are non-adjacent angles that lie on opposite sides of the transversal. In the figure above, the pairs of alternate interior angles are ∠1 and ∠3, ∠2 and ∠4. If two parallel lines are cut by a transversal, these pairs of angles have equal measurements.
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Co-Interior Angles: Co-interior angles are a pair of non-adjacent interior angles on the same side of the transversal. In the figure above, the pairs of co-interior angles are ∠1 and ∠4, ∠2 and ∠3. They are also known as same-side interior angles or consecutive interior angles. The sum of two co-interior angles is always 180º, which means they are also considered supplementary angles.
Interior Angles of a Triangle
A triangle has three interior angles at each vertex, and the sum of these interior angles is always 180°. Additionally, the bisectors of these angles meet at a point known as the incenter. The sum of the interior angles of a triangle being 180° means that each triangle can have only one possible right angle or obtuse angle. If all three interior angles are acute, the triangle is called an acute triangle. A triangle with one interior angle as obtuse is known as an obtuse triangle, while a triangle with one interior angle as a right angle is called a right-angled triangle.
Sum of Interior Angles Formula
From the simplest polygon, such as a triangle, to infinitely complex polygons with n sides like an octagon, the sum of the interior angles is a fascinating concept. The angle sum theorem tells us that multiplying two less than the number of sides by 180° gives us the sum of the interior angles in any polygon.
The formula for the sum of interior angles is:
Sum, S = (n − 2) × 180°
Here, S represents the sum of the interior angles, and n represents the number of sides of the polygon.
Let's apply this formula to a triangle:
S = (n − 2) × 180°
S = (3 − 2) × 180°
S = 1 × 180°
S = 180°
Using the same formula, we can calculate the sum of the interior angles of other polygons, as shown in the table below:
| Polygon | Number of sides, n | Sum of Interior Angles, S |
|---|---|---|
| Triangle | 3 | 180° |
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
| Heptagon | 7 | 900° |
| Octagon | 8 | 1080° |
| Nonagon | 9 | 1260° |
| Decagon | 10 | 1440° |
Finding an Unknown Interior Angle
The "Sum of Interior Angles Formula" can also be used to find an unknown interior angle of a polygon. Let's consider the following example to find the missing angle ∠x in the hexagon below:
According to the table above, the sum of the interior angles of a hexagon is 720°. Two of the interior angles in the above hexagon are right angles. Therefore, we can set up the following equation:
90 + 90 + 140 + 150 + 130 + x = 720°
Let's solve for x:
600 + x = 720
x = 720 - 600 = 120
Thus, the missing interior angle x is 120°.
Interior Angles of Regular Polygons
A polygon is considered regular when all its sides and angles are congruent. Regular polygons are captivating with their symmetrical beauty. Let's explore some examples:
We already know the formula for the sum of the interior angles of a polygon with 'n' sides is 180(n-2)°. Now, let's dive into the measurement of each interior angle in a regular polygon. Since all the interior angles of a regular polygon are equal, we can obtain the measurement of each angle by dividing the sum of the angles by the number of sides.
The formula for each interior angle is:
Each Interior Angle = ((180(n-2))/n)°
Let's apply this formula to find the interior angle of a regular pentagon. The number of sides of a pentagon is 5 (n = 5). Therefore, each interior angle of a regular pentagon can be found using the formula:
((180(n-2))/n)° = ((180(5-2))/5)° = (180 × 3)/5 = 540/5 = 108°
Thus, each interior angle of a regular pentagon measures 108°.
Using the same formula, we can calculate the interior angles of other regular polygons, as shown in the table below:
| Regular Polygon | Sum of Interior Angles, S | Measurement of Each Interior Angle ((180(n-2))/n)° |
|---|---|---|
| Triangle | 180° | 180/3 = 60°, Here n = 3 |
| Square | 360° | 360/4 = 90°, Here n = 4 |
| Pentagon | 540° | 540/5 = 108°, Here n = 5 |
| Hexagon | 720° | 720/6 = 120°, Here n = 6 |
| Heptagon | 900° | 900/7 = 128.57°, Here n = 7 |
| Octagon | 1080° | 1080/8 = 135°, Here n = 8 |
| Nonagon | 1260° | 1260/9 = 140°, Here n = 9 |
| Decagon | 1440° | 1440/10 = 144°, Here n = 10 |
Related Articles on Interior Angles
If you're interested in learning more about angles and geometry, check out the following related articles:
- Vertical Angles
- Alternate Angles
- Alternate Exterior Angles
- Same Side Interior Angles
- Interior Angles of Polygon Calculator
Final Words
As you delve into the world of interior angles, remember these important points:
- The sum of the interior angles of a polygon with 'n' sides can be calculated using the formula 180(n-2)°.
- Each interior angle of a regular polygon with 'n' sides can be calculated using the formula ((180(n-2))/n)°.
- The alternate interior angles theorem states that when a transversal intersects two parallel lines, each pair of alternate interior angles is equal. Conversely, if a pair of interior angles is equal when a transversal intersects two lines, then the two lines are parallel.
- The co-interior angles theorem states that if a transversal intersects two parallel lines, each pair of co-interior angles is supplementary (their sum is 180°). Conversely, if a pair of co-interior angles is supplementary, then the two lines are parallel.
Now that you have a solid understanding of interior angles, explore the fascinating world of geometry and its applications. Enjoy the journey!